Independent Brownian Motions Research Articles

Fully coupled forward backward stochastic differential equations driven by Lévy processes and application to differential games

Abstract We consider a fully coupled forward backward stochastic differential equation driven by a Lévy processes having moments of all orders and an independent Brownian motion. Under some monotonicity assumptions, we prove the existence and uniqueness of solutions on an arbitrarily fixed large time duration. We use this result to prove the existence of an open-loop Nash equilibrium point for non-zero sum stochastic differential games.

Slow kinetics of Brownian maxima.

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P(t) that the two maxima remain ordered up to time t and find the algebraic decay P ∼ t(-β) with exponent β = 1/4. When the two particles have diffusion constants D(1) and D(2), the exponent depends on the mobilities, β = (1/π) arctan sqrt[D(2)/D(1)]. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.

Central limit theorem for functionals of two independent fractional Brownian motions

We prove a central limit theorem for functionals of two independent d-dimensional fractional Brownian motions with the same Hurst index H in (2d+2,2d) using the method of moments.

Hunter, Cauchy rabbit, and optimal Kakeya sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Z n \mathbb {Z}_n . A hunter and a rabbit move on the nodes of Z n \mathbb {Z}_n without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al. (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first n n steps of order 1 / log ⁡ n 1/\log n . We show these strategies yield a Kakeya set consisting of 4 n 4n triangles with minimal area (up to constant), namely Θ ( 1 / log ⁡ n ) \Theta (1/\log n) . As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions { B ( s ) : s ≥ 0 } \{B(s): s \ge 0\} and { W ( s ) : s ≥ 0 } \{W(s): s \ge 0\} . Let τ t := min { s ≥ 0 : B ( s ) = t } \tau _t:=\min \{s \ge 0: B(s)=t\} . Then X t = W ( τ t ) X_t=W(\tau _t) is a Cauchy process and K := { ( a , X t + a t ) : a , t ∈ [ 0 , 1 ] } K:=\{(a,X_t+at) : a,t \in [0,1]\} is a Kakeya set of zero area. The area of the ε \varepsilon -neighbourhood of K K is as small as possible, i.e., almost surely of order Θ ( 1 / | log ⁡ ε | ) \Theta (1/|\log \varepsilon |) .

Multichannel deconvolution with long range dependence: Upper bounds on the [formula omitted]-risk [formula omitted

We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the Lp-risk (1≤p<∞). Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.

A note on the second-order non-autonomous neutral stochastic evolution equations with infinite delay under Carathéodory conditions

In this note, we aim to study a class of second-order non-autonomous neutral stochastic evolution equations with infinite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter H∈(1/2,1), in which the initial value belongs to the abstract space B. We establish the existence and uniqueness of mild solutions for this kind of equations under some Carathéodory conditions by means of the successive approximation. The obtained result extends some well-known results. An example is proposed to illustrate the theory.

Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

Perturbing the hexagonal circle packing: A percolation perspective

Nous considerons une juxtaposition hexagonale de cercles de rayon $1/2$ et nous la perturbons en laissant les cercles evoluer comme des mouvements browniens independants pendant un temps $t$. Nous montrons que, pour $t$ suffisamment grand, si $\varPi_{t}$ est le processus de points donne par les centres des cercles au temps $t$, alors quand $t\to\infty$, le rayon critique pour que les cercles centres en $\varPi_{t}$ contienne une composante infinie converge vers celui de la percolation continue (qui est strictement plus grand que $1/2$ comme l’ont montre Balister, Bollobas et Walters). D’un autre cote, pour $t$ suffisamment petit, nous montrons (a l’aide d’une estimation de Monte Carlo pour une integrale de grande dimension) que l’union des cercles contient une composante infinie. Nous discutons aussi des generalisations et des problemes ouverts.

Asymptotic behavior for bi-fractional regression models via Malliavin calculus

Let $$B^{H_1 ,K_1 }$$ and $$B^{H_2 ,K_2 }$$ be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence $$S_n : = \sum\limits_{i = 0}^{n - 1} {K\left( {n^\alpha B_i^{H_1 ,K_1 } } \right)\left( {B_{i + 1}^{H_2 ,K_2 } - B_i^{H_2 ,K_2 } } \right),}$$ where K is a standard Gaussian kernel function and the bandwidth parameter α satisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion $$B^{H_1 ,K_1 }$$ . We also give the stable convergence of the sequence S n by using the techniques of the Malliavin calculus.

Critical branching Brownian motion with absorption: survival probability

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $$-\sqrt{2}$$ . Kesten (Stoch Process 7:9–47, 1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $$t$$ . These bounds improve upon results of Kesten (Stoch Process 7:9–47, 1978), and partially confirm nonrigorous predictions of Derrida and Simon (EPL 78:60006, 2007).

Malliavin regularity of solutions to mixed stochastic differential equations

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.

Empirical Quantile Central Limit Theorems for Some Self-Similar Processes

In Swanson (Probab Theory Relat Fields 138:269–304, 2007), a central limit theorem (CLT) for the sample median of independent Brownian motions with value $$0$$ at $$0$$ was proved. Here, we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.

On time-dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay

In this paper, we show the existence and uniqueness of the mild solution for a class of time-dependent stochastic evolution equations with finite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter H ∈ (1 / 2,1). An example is provided to illustrate the theory. Copyright © 2013 John Wiley & Sons, Ltd.

First and Last Passage Times of Spectrally Positive Lévy Processes with Application to Reliability

We consider a wide class of increasing Levy processes perturbed by an independent Brownian motion as a degradation model. Such family contains almost all classical degradation models considered in the literature. Classically failure time associated to such model is defined as the hitting time or the first-passage time of a fixed level. Since sample paths are not in general increasing, we consider also the last-passage time as the failure time following a recent work by Barker and Newby (Reliab Eng Syst Saf 94:33–43, 2009). We address here the problem of determining the distribution of the first-passage time and of the last-passage time. In the last section we consider a maintenance policy for such models.

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let {W 1(t), t ∈ R +} and {W 2(t), t ∈ R +} be two independent Brownian motions with W 1(0) = W 2(0) = 0. {H(t) = W 1(|W 2(t)|), t ∈ R +} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets $$\{ t \in [0,T],H(t) = x\} $$ are established for any 0 < T ≤ 1.

Central limit theorem for a Stratonovich integral with Malliavin calculus

The purpose of this paper is to establish the convergence in law of the sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W), where W is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an It\^{o} integral of f''(W) with respect to a Gaussian martingale independent of W. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for q-fold Skorohod integrals, which is a multi-dimensional extension of a result proved by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39-64]. The results proved in this paper are generalizations of previous work by Swanson [Ann. Probab. 35 (2007) 2122-2159] and Nourdin and R\'{e}veillac [Ann. Probab. 37 (2009) 2200-2230], who found a similar formula for two particular types of bifractional Brownian motion. We provide three examples of Gaussian processes W that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters $H\le1/2$, HK=1/4. The others are Gaussian processes recently studied by Swanson [Probab. Theory Related Fields 138 (2007) 269-304], [Ann. Probab. 35 (2007) 2122-2159] in connection with the fluctuation of empirical quantiles of independent Brownian motion. In the first example the Gaussian martingale is a Brownian motion, and expressions are given for the other examples.

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.

Stochastic suppression and stabilization of nonlinear differential systems with general decay rate

In this paper, we investigate stochastic suppression and stabilization for a class of non-autonomous differential systems. Given a deterministic non-autonomous differential system, we introduce two independent Brownian motions and perturb this system into a new stochastic differential system. By using Lyapunov analysis method and some stochastic techniques, we show that a polynomial Brownian noise may guarantee the existence of global solution of the perturbed system while another linear Brownian noise may stabilize this system with general decay rate. An application of stochastic stabilization of differential system in the modeling of population growth is indicated.

A stochastic maximum principle in mean-field optimal control problems for jump diffusions

This paper is concerned with the study of a stochastic control problem, where the controlled system is described by a stochastic differential equation (SDE) driven by a Poisson random measure and an independent Brownian motion. The cost functional involves the mean of certain nonlinear functions of the state variable. The inclusion of this mean terms in the running and the final cost functions introduces a major difficulty when applying the dynamic programming principle. A key idea of solving the problem is to use the stochastic maximum principle method (SMP). In the first part of the paper, we focus on necessary optimality conditions while the control set is assumed to be convex. Then we prove that these conditions are in fact sufficient provided some convexity conditions are fulfilled. In the second part, the results are applied to solve the mean-variance portfolio selection problem in a jump setting.

Complex Brownian motion representation of the Dyson model

Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of $h$-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.